# Analysis Method to Deblurring Image

To deblur the image. usually we consider the model $$B=A(X)+e,$$ where $X$ is the expected image, $A$ is the convolution/matrix action and $B$ is the blurred image.

I know there are some matrix-methods to deal with the above case, e.g. in the book Deblurring Images: Matrices, Spectra, and Filtering.

Q: Are there some pde model/ functional model to understand the deblurring problem in mathematics. In other words, could anyone give some reference about the analytic methods to deblurr the image.

I'll assume the image to be recovered is an $m \times n$ array of real numbers. A very common method is to solve a convex optimization problem such as $$\operatorname{minimize}_{X \in \mathbb R^{m \times n}} \quad \frac12 \| A(X) - B \|^2 + \gamma \| Dx \|_1$$ where $D$ is a discrete gradient operator (in which case we're using "total variation regularization") or some sparsifying transformation such as a wavelet transformation. The norm $\| \cdot \|_1$ is just the usual $\ell_1$-norm, and $\| \cdot \|$ is the Frobenius norm.